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In our earlier works, we proposed rate adaptive hierarchical modulation-assisted two-best user opportunistic scheduling (TBS) and hybrid two-user scheduling (HTS) schemes. The proposed schemes are innovative in the sense that they include a second user in the transmission opportunistically using hierarchical modulations. As such the frequency of information access of the users increases without any degradation of the system spectral efficiency (SSE) compared to the classical opportunistic scheduling scheme. In this paper, we analyze channel access delay of an incoming packet at the base station (BS) buffer when our proposed TBS and HTS schemes are employed at the BS. Specifically, using a queuing analytic model we derive channel access delay as well as buffer distribution of the packets that wait at BS buffer for down-link (DL) transmission. We compare performance of the TBS and HTS schemes with that of the classical single user opportunistic schemes namely, absolute carrier-to-noise ratio (CNR)-based single user scheduling (ASS) and normalized CNR-based single user scheduling (NSS). For an independent and identically distributed (i.i.d.) fading environment, our proposed scheme can improve packet's access delay performance compared to the ASS. Selected numerical results in an independent but non-identically distributed (i.n.d.) fading environment show that our proposed HTS achieves overall good channel access delay performance.